Edges not in any monochromatic copy of a fixed graph

Hong Liu Oleg Pikhurko Maryam Sharifzadeh ∗

July 10, 2018

Abstract

For a sequence (Hi)ki=1 of graphs, let nim(n;H1, . . . ,Hk) denote the maximum number of

edges not contained in any monochromatic copy of Hi in colour i, for any colour i, over allk-edge-colourings of Kn.

When each Hi is connected and non-bipartite, we introduce a variant of Ramsey numberthat determines the limit of nim(n;H1, . . . ,Hk)/

(n2

)as n → ∞ and prove the corresponding

stability result. Furthermore, if each Hi is what we call homomorphism-critical (in particularif each Hi is a clique), then we determine nim(n;H1, . . . ,Hk) exactly for all sufficiently large n.The special case nim(n;K3,K3,K3) of our result answers a question of Ma.

For bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., whenk = 2 and H1 = H2). It is trivial to see that nim(n;H,H) is at least ex(n,H), the maximumsize of an H-free graph on n vertices. Keevash and Sudakov showed that equality holds if H isthe 4-cycle and n is large; recently Ma extended their result to an infinite family of bipartitegraphs. We provide a larger family of bipartite graphs for which nim(n;H,H) = ex(n,H). Fora general bipartite graph H, we show that nim(n;H,H) is always within a constant additiveerror from ex(n,H), i.e., nim(n;H,H) = ex(n,H) +OH(1).

1 Introduction

Many problems of extremal graph theory ask for (best possible) conditions that guarantee theexistence of a given ‘forbidden’ graph. Two prominent examples of this kind are the Turan functionand Ramsey numbers. Recall that, for a graph H and an integer n, the Turan function ex(n,H) isthe maximum size of an n-vertex H-free graph. Let Kt denote the complete graph on t vertices. Thefamous theorem of Turan [48] states that the unique maximum Kr+1-free graph of order n is theTuran graph T (n, r), the complete balanced r-partite graph. Thus ex(n,Kk+1) = t(n, r), where wedenote t(n, r) := e(T (n, r)). For a sequence a1, . . . , ak of integers, the Ramsey number R(a1, . . . , ak)is the minimum R such that for every edge-colouring of KR with colours from [k] := {1, . . . , k}, thereis a colour-i copy of Kai for some i ∈ [k]. The fact that R exists (i.e., is finite) was first establishedby Ramsey [39] and then independently rediscovered by Erdos and Szekeres [16]. Both of theseproblems motivated a tremendous amount of research, see e.g. the recent surveys by Conlon, Foxand Sudakov [4], Furedi and Simonovits [20], Keevash [28], Radziszowski [38] and Sudakov [44].

∗Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, UK. Email addresses: {h.liu.9,o.pikhurko, m.sharifzadeh}@warwick.ac.uk. H.L. was supported by the Leverhulme Trust Early Career Fellow-ship ECF-2016-523. O.P. was supported by ERC grant 306493 and EPSRC grant EP/K012045/1. M.Sh. wassupported by ERC grant 306493 and Marie Curie Individual Fellowship 752426.

1

A far-reaching generalisation is to ask for the number of guaranteed forbidden subgraphs. Forthe Turan function this gives the famous Erdos-Rademacher problem that goes back to Rademacher(1941; unpublished): what is the minimum number of copies of H in a graph of given order n andsize m > ex(n,H)? This problem was revived by Erdos [8, 9] in the 1950–60s. Since then itcontinues to be a very active area of research, for some recent results see e.g. [3, 5, 6, 26, 30, 32, 34,35, 36, 40, 41, 45]. The analogous question for Ramsey numbers, known as the Ramsey multiplicityproblem, was introduced by Erdos [10] in 1962 and is wide open, see e.g. [2, 7, 17, 21, 27, 43, 46, 47].

A less studied but still quite natural question is to maximise the number of edges that do notbelong to any forbidden subgraph. Such problems in the Turan context (where we are given theorder n and the size m > ex(n,H) of a graph G) were studied in [13, 19, 22, 23]. In the Ramseycontext, a problem of this type seems to have been first posed by Erdos, Rousseau, and Schelp(see [12, Page 84]). Namely, they considered the maximum number of edges not contained in anymonochromatic triangle in a 2-edge-colouring of Kn. Also, Erdos [12, Page 84] wrote that “manyfurther related questions can be asked”. Such questions will be the focus of this paper.

Let us provide a rather general definition. Suppose that we have fixed a sequence of graphsH1, . . . ,Hk. For a k-edge-colouring φ of Kn, let NIM(φ) consist of all NIM-edges, that is, thoseedges of Kn that are not contained in any colour-i copy of Hi for any i ∈ [k]. In other words,NIM(φ) is the complement (with respect to E(Kn)) of the union over i ∈ [k] of the edge-sets ofHi-subgraphs of colour-i. Define

nim(n;H1, . . . ,Hk) := maxφ:E(Kn)→[k]

|NIM(φ)|,

to be the maximum possible number of NIM-edges in a k-edge-colouring of Kn. If all Hi’s are thesame graph H, we will write nimk(n;H) instead. Note that for k = 2 by taking one colour-class tobe a maximum H-free graph, we have nim2(n;H) ≥ ex(n,H). In ([12, Page 84]), Erdos mentionedthat together with Rousseau and Schelp, they showed that in fact

nim2(n;H) = ex(n,H), for all n ≥ n0(H), (1)

when H = K3 is the triangle. As observed by Alon (see [29, Page 42]), this also follows froman earlier paper of Pyber [37]. Keevash and Sudakov [29] showed that (1) holds when H is anarbitrary clique Kt (or, more generally, when H is edge-colour-critical, that is, the removal of someedge e ∈ E(H) decreases the chromatic number) as well as when H is the 4-cycle C4 (and n ≥ 7).They [29, Problem 5.1] also posed the following problem.

Problem 1.1 (Keevash and Sudakov [29]). Does (1) hold for every graph H?

In a recent paper, Ma [33] answered Problem 1.1 in the affirmative for the infinite family ofreducible bipartite graphs, where a bipartite graph H is called reducible if it contains a vertexv ∈ V (H) such that H − v is connected and ex(n,H − v) = o(ex(n,H)) as n → ∞. Ma [33] alsostudied the case of k ≥ 3 colours and raised the following question.

Question 1.2 (Ma [33]). Is it true that nim3(n;K3) = t(n, 5)?

The lower bound in Question 1.2 follows by taking a blow-up of a 2-edge-colouring of K5 withouta monochromatic triangle, and assigning the third colour to all pairs inside a part.

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1.1 Non-bipartite case

In order to state some of our results, we have to introduce the following variant of Ramsey number.Given a set X, denote by

(Xi

)(resp.

(X≤i)), the set of all subsets of X of size i (resp. at most i).

Given two graphs H and G, a (not necessarily injective) map φ : V (H)→ V (G) is a homomorphismif it preserves all adjacencies, i.e. φ(u)φ(v) ∈ E(G) for every uv ∈ E(H), and we say that G is ahomomorphic copy of H.

Definition 1.3. Given a sequence of graphs H = (H1, . . . ,Hk), denote by r∗(H1, . . . ,Hk) the

maximum integer r∗ such that there exists a colouring ξ :([r∗]≤2)→ [k] such that

(P1) the restriction of ξ to([r∗]2

)is (H1, . . . ,Hk)-homomorphic-free (that is, for each i ∈ [k] there

is no edge-monochromatic homomorphic copy of Hi in the i-th colour);

(P2) for every distinct i, j ∈ [r∗] we have ξ({i, j}) 6= ξ({i}), that is, we forbid a pair having thesame colour as one of its points.

For any r′ ≤ r∗, we will call a colouring ξ :([r′]≤2)→ [k] feasible (with respect to (H1, . . . ,Hk))

if it satisfies both (P1) and (P2). We say that (H1, . . . ,Hk) is nice if every feasible colouring

ξ :([r∗(H1,...,Hk)]

≤2)→ [k] assigns the same colour to all singletons.

Note that the colour assigned by ξ to the empty set ∅ ∈([r∗]≤2)

does not matter. Note also thatwhen k = 2, due to (P2), a feasible colouring should use the same colour on all singletons. Thus,r∗(H1, H2) = max{χ(H1), χ(H2)} − 1. If we ignore (P2), then we obtain the following variantof Ramsey number that was introduced by Burr, Erdos and Lovasz [1]. Let rhom(H1, . . . ,Hk)be the homomorphic-Ramsey number, that is the maximum integer r such that there exists an(H1, . . . ,Hk)-homomorphic-free colouring ξ :

([r]2

)→ [k]. We remark that for the homomorphic-

Ramsey number, the colours of vertices do not play a role. When all Hi’s are cliques, this Ramseyvariant reduces to the classical graph Ramsey problem:

rhom(Ka1 , . . . ,Kak) = R(a1, . . . , ak)− 1. (2)

There are some further relations to r∗. For example, by assigning the same colour i to allsingletons and using the remaining k − 1 colours on pairs, one can see that

r∗(H1, . . . ,Hk) ≥ maxi∈[k]

rhom(H1, . . . ,Hi−1, Hi+1, . . . ,Hk). (3)

If some Hi is bipartite, then the problem of r∗ reduces to rhom. Indeed, as K2 is a homomorphiccopy of any bipartite graph, when some Hi is bipartite, no feasible colouring ξ can use colour ion any pair. Consequently, we have equality in (3). This is one of the reasons why we restrict tonon-bipartite Hi in this section.

It would be interesting to know if (3) can be strict. We conjecture that if all Hi’s are cliquesthen there is equality in (3) and, furthermore, every optimal colouring uses the same colour on allsingletons:

Conjecture 1.4. For any integers 3 ≤ a1 ≤ . . . ≤ ak, (Ka1 , . . . ,Kak) is nice. In particular,r∗(Ka1 , . . . ,Kak) = R(Ka2 , . . . ,Kak)− 1.

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Figure 1: A feasible colouring of K4 with respect to (C5, C5, C5), with two different colours onvertices.

It is worth noting that not all k-tuples are nice. For example, it is easy to show that r∗(C5, C5, C5) =rhom(C5, C5) = 4, where Ci denotes the cycle of length i, while Figure 1 shows a feasible colouring

of( [4]≤2)

assigning two different colours to singletons.Our first result shows that this new variant plays a similar role for the function nim(·) as the

chromatic number in the Erdos-Simonovits-Stone Theorem [15, 14].

Theorem 1.5. Let Hi be a non-bipartite graph, i ∈ [k], and let r∗ := r∗(H1, . . . ,Hk). For everyε > 0, we have that, for all large n,

nim(n;H1, . . . ,Hk) ≤(

1− 1

r∗

)n2

2+ εn2. (4)

Furthermore, if each Hi is connected or there exists a feasible colouring of([r∗]≤2)

with k colours suchthat all singletons have the same colour, then we have nim(n;H1, . . . ,Hk) ≥ t(n, r∗).

We also obtain the following stability result stating that if the number of NIM-edges is closeto the bound in (4), then the NIM-graph is close to a Turan graph. Let the edit distance betweengraphs G and H of the same order be

δedit(G,H) := minσ|E(G)4 σ(E(H))|, (5)

where the minimum is taken over all bijections σ : V (H) → V (G). In other words, δedit(G,H) isthe minimum number of adjacency edits needed to make G and H isomorphic.

Theorem 1.6. For any non-bipartite graphs Hi, i ∈ [k], and any constant ε > 0, there existsδ > 0 such that the following holds for sufficiently large n. If the number of NIM-edges of someφ :([n]2

)→ [k] satisfies

nim(φ;H1, . . . ,Hk) ≥(

1− 1

r∗

)n2

2− δn2,

then δedit(Gnim, T (n, r∗)) ≤ εn2, where r∗ := r∗(H1, . . . ,Hk) and Gnim is the NIM-graph of φ,

i.e., the spanning subgraph with edge set NIM(φ).

Our next theorem shows that if Conjecture 1.4 is true, then this would determine the exact valueof nim(·) for a rather large family of graphs, including cliques. We call a graph H homomorphism-critical if it satisfies the following. If F is a minimal homomorphic copy ofH, i.e. no proper subgraphof F is a homomorphic copy of H, then for any edge uv ∈ E(F ), there exists a homomorphismg : V (H)→ V (F ) such that |g−1(u)| = |g−1(v)| = 1, i.e. the pre-image sets of u and v are singletons.

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For example, all complete multipartite graphs with at least two parts of size 1 are homomorphism-critical. A simple consequence of this property is the following. As F is minimal, it does nothave any isolated vertices. Therefore, for any vertex v ∈ V (F ), there exists a homomorphismg : V (H)→ V (F ) such that |g−1(v)| = 1.

Theorem 1.7. Let (H1, . . . ,Hk) be a nice sequence of non-bipartite graphs such that each Hi ishomomorphism-critical. Then for sufficiently large n,

nim(n;H1, . . . ,Hk) = t(n, r∗),

where r∗ := r∗(H1, . . . ,Hk). Additionally, the NIM-graph of every extremal colouring is isomorphicto T (n, r∗).

In the following theorems, we prove Conjecture 1.4 for k = 3, and for a1 = . . . = a4 = 3 whenk = 4.

Theorem 1.8. For all integers 3 ≤ a1 ≤ a2 ≤ a3, (Ka1 ,Ka2 ,Ka3) is nice. In particular,

r∗(Ka1 ,Ka2 ,Ka3) = R(a2, a3)− 1.

Theorem 1.9. We have that (K3,K3,K3,K3) is nice. In particular,

r∗(K3,K3,K3,K3) = R(3, 3, 3)− 1 = 16.

The following is an immediate corollary of Theorems 1.7, 1.8 and 1.9. In particular, the specialcase a1 = a2 = a3 = 3 answers Question 1.2 affirmatively.

Corollary 1.10. Let 3 ≤ a1 ≤ a2 ≤ a3 be integers. Then for sufficiently large n,

nim(n;Ka1 ,Ka2 ,Ka3) = t(n,R(a2, a3)− 1),

nim4(n;K3) = t(n, 16), and the NIM-graph of every extremal colouring is the corresponding Turangraph.

1.2 Bipartite graphs

For bipartite graphs, we will provide a new family for which Problem 1.1 has a positive answer.Let us call an h-vertex graph H weakly-reducible if there exist n0 ∈ N and a vertex v ∈ V (H) suchthat ex(n,H − v) < ex(n,H)− 22h

2n for all n ≥ n0. (The function 22h

2comes from the proof and

we make no attempt to optimise it.) Note that the family of weakly-reducible graphs includes allreducible graphs except the path of length 2 and this inclusion is strict. For example, for integerst > s ≥ 2, the disjoint union of the complete bipartite graphs K2,t and K2,s is weakly-reduciblebut not reducible; this can be easily deduced from the result of Furedi [18] that ex(n,K2,k) =(√k/2 + o(1))n3/2 for any fixed k ≥ 2 as n→∞.

Theorem 1.11. Let H be a weakly-reducible bipartite graph and n be sufficiently large. Then

nim2(n;H) = ex(n,H).

Furthermore, every extremal colouring has one of its colour classes isomorphic to a maximumH-free graph of order n.

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For a general bipartite graph H, we give in the following two theorems a weaker bound with anadditive constant error term, namely,

nim2(n;H) ≤ ex(n,H) +OH(1).

This provides more evidence towards Problem 1.1.

Theorem 1.12. Let H be a bipartite graph on at most h vertices containing at least one cycle.Then for sufficiently large n,

nim2(n;H) ≤ ex(n,H) + h2.

For more than 2 colours, we obtain an asymptotic result for trees. Fix a tree T , by taking randomoverlays of k−1 copies of extremal T -free graphs, we see that nimk(n;T ) ≥ (k−1)ex(n, T )−Ok,|T |(1)(this construction is from Ma [33]). We prove that this lower bound is asymptotically true.

Theorem 1.13. Let T be a forest with h vertices. If k = 2 or if T is a tree, then there exists aconstant C := C(k, h) such that, for all sufficiently large n,∣∣nimk(n;T )− (k − 1) ex(n, T )

∣∣ ≤ C(k, h).

Organisation of the paper. We first introduce some tools in Section 2. Then in Section 3, wewill prove Theorems 1.11, 1.12, and 1.13. In Section 4, we will prove Theorems 1.5 and 1.6. We willpresent the proof for Theorem 1.7 in Section 5 and the proofs of Theorems 1.8 and 1.9 in Section 6.Finally, in Section 7 we give some concluding remarks.

2 Preliminaries

In this section, we recall and introduce some notation and tools. Recall that [m] := {1, 2, . . . ,m}and

(Xi

)(resp.

(X≤i)) denotes the set of all subsets of a set X of size i (resp. at most i). We also

use the term i-set for a set of size i. We may abbreviate a singleton {x} (resp. a pair {x, y}) as x(resp. xy). If we claim, for example, that a result holds whenever 1� a� b > 0, this means thatthere are a constant a0 ∈ (0, 1) and a non-decreasing function f : (0, 1)→ (0, 1) (that may dependon any previously defined constants or functions) such that the result holds for all a, b ∈ (0, 1) witha ≤ a0 and b ≤ f(a). We may omit floors and ceilings when they are not essential.

Let G = (V,E) be a graph. Its order is v(G) := |V | and its size is e(G) := |E|. The complement

of G is G :=(V,(V2

)\ E)

. The chromatic number of G is denoted by χ(G). For U ⊆ V , let

G[U ] := (U, {xy ∈ E : x, y ∈ U}) denote the subgraph of G induced by U . Also, denote

NG(v, U) := {u ∈ U | uv ∈ E},dG(v, U) := |NG(v, U)|,

and abbreviate NG(v) := NG(v, V ) and dG(v) := dG(v, V ). Let δ(G) := min{dG(v) : v ∈ V } denotethe minimum degree of G.

Let U = {U1, U2, . . . , Uk} be a collection of disjoint subsets of V . We write G[U1, . . . , Uk] or G[U ]for the multipartite subgraph of G with vertex set U := ∪i∈[k]Ui where we keep the cross-edges ofG (i.e. edges that connect two parts); equivalently, we remove all edges from G[U ] that lie inside a

6

part Ui ∈ U . In these shorthands, we may omit G whenever it is clear from the context, e.g. writing[U1, . . . , Uk] for G[U1, . . . , Uk]. We say that U is a max-cut k-partition of G if e(G[U1, . . . , Uk]) ismaximised over all k-partitions of V (G).

For disjoint sets V1, . . . , Vt with t ≥ 2, let K[V1, . . . , Vt] denote the complete t-partite graphwith parts V1, . . . , Vt. Its isomorphism class is denoted by K|V1|,...,|Vt|. For example, if part sizesdiffer by at most 1, then we get the Turan graph T (|V1|+ . . .+ |Vt|, t). Let Mh denote the matchingwith h edges.

Definition 2.1. For an edge-colouring φ :([n]2

)→ [k] of G := Kn, define NIM(φ;H1, . . . ,Hk)

to be the set of all edges not contained in any monochromatic copy of Hi in colour i, and letnim(φ;H1, . . . ,Hk) := |NIM(φ;H1, . . . ,Hk)|. Thus

nim(n;H1, . . . ,Hk) = maxφ:E(Kn)→[k]

nim(φ;H1, . . . ,Hk).

If the Hi’s are all the same graph H, then we will use the shorthands NIMk(φ;H), nimk(φ;H)and nimk(n;H) respectively. Also, we may drop k when k = 2 and omit the graphs Hi when theseare understood. Let Gnim be the spanning subgraph of G with E(Gnim) = NIM(φ;H1, . . . ,Hk). Fori ∈ [k], denote by Gi and Gnim

i the spanning subgraphs of G with edge-sets E(Gi) = {e ∈ E(G) :φ(e) = i} and E(Gnim

i ) = {e ∈ E(Gnim) : φ(e) = i}. We call an edge e ∈ E(Gnim) (respectively,e ∈ E(Gnim

i )) a NIM-edge (resp. a NIM-i-edge).

Definition 2.2. For ξ :( [t]≤2)→ [k] and disjoint sets V1, . . . , Vt, the blow-up colouring ξ(V1, . . . , Vt) :(

V1∪···∪Vt2

)→ [k] is defined by

ξ(V1, . . . , Vt)(xy) :=

{ξ(ij), if xy ∈ E(K[Vi, Vj ]),

ξ(i), if x, y ∈ Vi.

If |Vi| = N for every i ∈ [t], then we say that ξ(V1, . . . , Vt) is an N -blow-up of ξ.

We say that a colouring φ contains another colouring ψ and denote this by φ ⊇ ψ if ψ isa restriction of φ. In particular, φ ⊇ ξ(V1, . . . , Vt) means that φ is defined on every pair insideV1 ∪ · · · ∪ Vt and coincides with ξ(V1, . . . , Vt) there.

We will make use of the multicolour version of the Szemeredi Regularity Lemma (see, forexample, [31, Theorem 1.18]). Let us recall first the relevant definitions. Let X,Y ⊆ V (G) bedisjoint non-empty sets of vertices in a graph G. The density of (X,Y ) is

d(X,Y ) :=e(G[X,Y ])

|X| |Y |.

For ε > 0, the pair (X,Y ) is ε-regular if for every pair of subsets X ′ ⊆ X and Y ′ ⊆ Y with|X ′| ≥ ε|X| and |Y ′| ≥ ε|Y |, we have |d(X,Y ) − d(X ′, Y ′)| ≤ ε. Additionally, if d(X,Y ) ≥ γ,for some γ > 0, we say that (X,Y ) is (ε, γ)-regular. A partition P = {V1, . . . , Vm} of V (G) is anε-regular partition of a k-edge-coloured graph G if

1. for all ij ∈([m]2

),∣∣ |Vi| − |Vj | ∣∣ ≤ 1;

2. for all but at most ε(m2

)choices of ij ∈

([m]2

), the pair (Vi, Vj) is ε-regular in each colour.

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Lemma 2.3 (Multicolour Regularity Lemma). For every real ε > 0 and integers k ≥ 1 and M ,there exists M ′ such that every k-edge-coloured graph G with n ≥ M vertices admits an ε-regularpartition V (G) = V1 ∪ . . . ∪ Vr with M ≤ r ≤M ′.

Given ε, γ > 0, a graph G, a colouring φ : E(G) → [k] and a partition V (G) = V1 ∪ · · · ∪ Vr,define the reduced graph

R := R(ε, γ, φ, (Vi)ri=1) (6)

as follows: V (R) = {V1, . . . , Vr} and Vi and Vj are adjacent in R if (Vi, Vj) is ε-regular with respectto the colour-` subgraph of G for every ` ∈ [k] and the colour-m density of (Vi, Vj) is at least γfor some m ∈ [k]. For brevity, we may omit φ or (Vi)

ri=1 in (6) when these are clear. The graph

R comes with the majority edge-colouring which assigns to each edge ViVj ∈ E(R) the colour thatis the most common one among the edges in G[Vi, Vj ] under the colouring φ. In particular, themajority colour has density at least γ in G[Vi, Vj ]. We will use the following consequence of theEmbedding Lemma (see e.g. [31, Theorem 2.1]).

Lemma 2.4 (Emdedding Lemma). Let H and R be graphs and let 1 ≥ γ � ε� 1/m > 0. Let Gbe a graph obtained by replacing every vertex of R by m vertices, and replacing the edges of R withε-regular pairs of density at least γ. If R contains a homomorphic copy of H, then H ⊆ G.

We will also need the Slicing Lemma (see e.g. [31, Fact 1.5]).

Lemma 2.5 (Slicing Lemma). Let ε, α, γ ∈ (0, 1) satisfy ε ≤ min{γ, α, 1/2}. If (A,B) is an (ε, γ)-regular pair, then for any A′ ⊆ A and B′ ⊆ B with |A′| ≥ α|A| and |B′| ≥ α|B|, we have that(A′, B′) is an (ε′, γ − ε)-regular pair, where ε′ := max{ε/α, 2ε}.

Conventions: Throughout the rest of this paper, we will use G as an edge-coloured Kn. For agiven number of colours k and a sequence of graphs (Hi)

ki=1, we will always write ψ :

([n]2

)→ [k] for

an extremal colouring realising nim(n;H1, . . . ,Hk). We do not try to optimise the constants norprove most general results, instead aiming for the clarify of exposition.

3 Proofs of Theorems 1.11, 1.12 and 1.13

By adding isolated vertices, we can assume that each graph Hi has even order. The followingproposition will be frequently used. It basically says that there are no monochromatic copies ofKv(Hi),v(Hi)/2 in colour i that contains a NIM-i-edge. Its proof follows from the fact that every edgeof Kv(Hi),v(Hi)/2 is in an Hi-subgraph.

Proposition 3.1. For every graph G, fixed bipartite graphs H1, . . . ,Hk, and a k-edge-colouringφ : E(G) → [k], we have the following for every vertex v ∈ V (G) and i ∈ [k]. Let Ui := {v′ ∈V (G) : vv′ ∈ Gnim

i }.

(i) For every vertex u ∈ Ui, the graph Gi[NGi(v) \ {u}, NGi(u) \ {v}] is Kv(Hi),v(Hi)/2-free.

(ii) The graph Gi[Ui, V \ (Ui ∪ {v})] is Kv(Hi),v(Hi)/2-free.

One of the key ingredients for the 2-colour case for bipartite graphs is the following lemma,which is proved by extending an averaging argument of Ma [33]. It states that any 2-edge-colouringof Kn has only linearly many NIM-edges, or there is neither large NIM star nor matching in one ofthe colours.

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Lemma 3.2. For any h-vertex bipartite H with h even and any 2-edge-colouring φ of G := Kn

with nim(φ;H) > 22h2n, there exists i ∈ [2] such that Gnim

i is {K1,h,Mh/2}-free.

Proof. We may assume, without loss of generality, that Gnim1 contains K1,h, since otherwise

nim(φ;H) ≤ 2 · ex(n,K1,h) ≤ (h− 1)n,

contradicting nim(φ;H) > 22h2n. Let Sv be an h-star in Gnim

1 centred at v. We will show that ifGnim

2 contains the starK1,h (Case 1) or the matchingMh/2 (Case 2), then it follows that nim(φ;H) ≤22h

2n, which is a contradiction. In each case, we will define a set S ⊆ V (G), with h+ 1 ≤ |S| ≤ h2,

containing Sv as follows. In Case 1, let Su be an h-star centred at u in Gnim2 (u and v are not

necessarily distinct). Define S = V (Sv) ∪ V (Su) with h + 1 ≤ |S| ≤ 2h + 2. In Case 2, letM ⊆ Gnim

2 be a matching with edge set {e1, . . . , eh/2}, where ei = zi,1zi,2 for every 1 ≤ i ≤ h/2.

Denote Z := ∪h/2i=1{zi,1, zi,2}. For each edge ei ∈ E(M), without loss of generality, assume thatdG2(zi,1) ≥ dG2(zi,2). Define iteratively for every i = 1, . . . , h/2 a set U ′i as follows,{

U ′i ⊆Wi, |U ′i | = h2 , if |Wi| ≥ h/2,

U ′i = Wi, otherwise,

where Wi := NG2(zi,1) \(Z ∪

(∪i−1j=1U

′j

)); further define Ui := U ′i ∪ {zi,1, zi,2}. Finally, set S :=(

∪h/2i=1Ui

)∪ V (Sv). So h+ 1 ≤ |S| ≤ h+ 1 + (h/2 + 2) · h/2 ≤ h2.

We now define a partition of V (G) \ S that will be used in both Case 1 and Case 2. For eachvertex w ∈ V (G) \ S, denote by fw the function S → [2] whose value on s ∈ S is fw(s) = φ(sw).In other words, fw encodes the colours of the edges from w to S. Define

Y1 := { v ∈ V (G) \ S : |f−1v (2)| < h/2 },Y2 := { v ∈ V (G) \ S : |f−1v (1)| < h/2 },X := V (G) \ (S ∪ Y1 ∪ Y2).

Thus X consists of those v ∈ V (G) \ S that send at least h/2 edges of each colour to S.We will show in the following claims that, for each class in this partition, there are few vertices

in that class or the number of NIM-edges incident to it is linear.

Claim 3.3. e(Gnim[X]) ≤ h( |S|h/2

)n.

Proof of Claim. Assume to the contrary that e(Gnimi [X]) ≥ h

( |S|h/2

)n/2, for some i ∈ [2]. Then

there exists a vertex x ∈ X with dGnimi [X](x) ≥ h

( |S|h/2

). By the definition of X, each vertex in

NGnimi [X](x) has at least h/2 Gi-neighbours in S. By the Pigeonhole Principle, there exists a copy

of Kh,h/2 ⊆ Gi[NGnimi [X](x), S], which is a contradiction by Proposition 3.1(ii).

Claim 3.4. |Y1| < h · 2|S|.

Proof of Claim. Assume to the contrary that |Y1| ≥ h · 2|S|. Since the total number of functionsS → [2] is 2|S|, by averaging, there exists a function f and a subset Yf ⊆ Y1 with |Yf | ≥ h suchthat for all vertices y ∈ Yf , the functions f and fy are the same. By the definition of Y1, there is a

9

subset I ⊆ V (Sv) \ {v} with |I| ≥ h/2 such that for all s ∈ I, f(s) = 1, i.e., all pairs between Yfand I are of colour 1. Recall that Sv is the h-star consisting of NIM-i-edges, thus, there exists acopy of Kh/2,h ⊆ G1[NGnim

1(v), Yf ], which contradicts Proposition 3.1(ii).

We now show that Y2 has also to be small (given that Gnim2 contains a large star or matching),

otherwise nim(φ,H) is linear.

Case 1: Gnim2 has the star K1,h.

A similar argument as in Claim 3.4 (with Su playing the role of Sv) shows that |Y2| < h · 2|S|.

Case 2: Gnim2 has the matching Mh/2.

By the definition of S, all the Ui’s are pairwise disjoint and h+1 ≤ |S| ≤ h2, see Figure 2. Suppose

XY2Y1

U0� U

0h=2

z���

z��2

zh=2��

zh=2�2

jU�j �h2� 2 jUh=2j �

h2� 2

�

v

�v

Figure 2: Case 2 of Lemma 3.2

that |Y2| ≥ h · 2|S|. Again there exists a function f : S → [2] with |f−1(1)| < h/2 and a subsetYf ⊆ Y2 with |Yf | ≥ h, such that for all vertices y ∈ Yf , fy is the same as f . We will use thefollowing claim.

Claim 3.5. For every 1 ≤ i ≤ h/2, there exists w ∈ Ui such that f(w) = 1.

Proof of Claim. For a fixed 1 ≤ i ≤ h/2, assume to the contrary that for all s ∈ Ui, we havef(s) = 2, i.e., E(G[Ui, Yf ]) ⊆ E(G2). Thus, |NG2(zi,1) \ (∪i−1j=1U

′j ∪ Z)| ≥ |Yf |. Consequently,

|U ′i | = h/2. Therefore, there exists Kh/2,h ⊆ G2[Ui \{zi,1}, Yf ], which contradicts Proposition 3.1(i)with zi,1 and zi,2 playing the roles of v and u respectively.

By Claim 3.5 together with the fact that the Ui’s are pairwise disjoint, f assumes value 1 atleast h/2 times, which contradicts Yf ⊆ Y2. Therefore, in both cases, |Y2| < h · 2|S|.

Let Y := Y1 ∪ Y2. Since |S| ≤ h2, by Claims 3.3 and 3.4, we get that

nim(φ;H) ≤ e(Gnim[S]) + e(Gnim[S, V \ S]) + e(Gnim[Y ]) + e(Gnim[Y,X]) + e(Gnim[X])

≤ |S| · n+ |Y | · n+ e(Gnim[X])

≤ (h2 + 2 · h2h2)n+ h

(h2

h/2

)n < 22h

2n,

a contradiction.This completes the proof of Lemma 3.2.

10

3.1 Weakly-reducible bipartite graphs

Proof of Theorem 1.11. Let H be a weakly-reducible bipartite graph. Let h = v(H) and w ∈ V (H)be a vertex such that ex(n,H − w) < ex(n,H) − 22h

2n for n ≥ n0. In particular, we have that

ex(n,H) > 22h2n for n ≥ n0. Thus by Lemma 3.2, we may assume that there is i ∈ [2] such that

e(Gnimi ) ≤ ex(n, {K1,h,Mh/2}) ≤ h2. By the symmetry between the two colours, let us assume

that i = 1. Suppose that E(Gnim1 ) 6= ∅ as otherwise we are trivially done. We now distinguish the

following two cases.

Case 1: For every edge e = uv ∈ E(Gnim1 ), dG1(u) ≤ 10h and dG1(v) ≤ 10h.

In this case, pick one such edge, e = uv, and define V1 = (NG1(u) ∪NG1(v)) \ {u, v}. So |V1| ≤dG1(u) + dG1(v) ≤ 20h. Let

V2 := V (G) \ (V1 ∪ {u, v}) = NG2(u) ∩NG2(v).

Note that the subgraph of Gnim2 induced on vertex set V2 satisfies e(Gnim

2 [V2]) ≤ ex(n,H − w).Otherwise, a copy of H −w in Gnim

2 [V2] together with u forms a copy of H in colour 2. Recall that|V1| ≤ 20h, e(Gnim

1 ) ≤ h2 and V1∪V2∪{u, v} is a partition of V (G). Therefore for large n, we have

nim(ψ;H) ≤ e(Gnim2 ) + e(Gnim

1 ) ≤ e(Gnim2 [V2]) + (|V1|+ 2)n+ h2 (7)

≤ ex(n,H − w) + 30hn ≤ ex(n,H)− 22h2n+ 30hn < ex(n,H).

Case 2: There exists an edge e = uv ∈ E(Gnim1 ) such that dG1(u) ≥ 10h.

Pick A ⊆ NG1(u) with |A| = 10h, and denote

X := {z ∈ V (G) \ (A ∪ {u, v}) : dG2(z,A) ≥ h},Y := {z ∈ V (G) \ (A ∪ {u, v}) : dG1(z,A) ≥ h} \X.

Note that X ∪ Y ∪A ∪ {u, v} is a partition of V (G). We will use the following claims.

Claim 3.6. For every vertex w ∈ X ∪ Y , dGnim2

(w,X) < h(10hh

).

Proof of Claim. Assume to the contrary that there exists a vertex w ∈ X ∪ Y with dGnim2

(w,X) ≥h(10hh

), and define S := NGnim

2(w,X). Since |A| = 10h and vertices in S all have G2-degree at least

h in A, there exists a subset of S of size at least h such that its vertices are connected in G2 to thesame h vertices in A, i.e., Kh,h ⊆ G2[S,A], which contradicts Proposition 3.1(ii).

Define Y ′ = Y ∩NG1(v) to be the set of all vertices in Y that are adjacent to v with a 1-colourededge, and Y ′′ = Y \ Y ′.

Claim 3.7. |Y ′| <(10hh

)h.

Proof of Claim. Assume to the contrary that |Y ′| ≥(10hh

)h. Since all vertices in Y ′ have at least h

G1-neighbours in A, there exists a copy of Kh,h ⊆ G1[Y′, A], which extends to a copy of Kh+1,h+1 ⊇

H containing the edge uv ∈ E(Gnim1 ), a contradiction.

11

By Claims 3.6, 3.7 and since |A| = 10h, the number of edges in Gnim2 with at least one end point

in the set A ∪ Y ′ ∪X ∪ {u, v} is at most 3h(10hh

)n. It remains to estimate e(Gnim

2 [Y ′′]). We claimthat e(Gnim

2 [Y ′′]) ≤ ex(n,H −w). Otherwise, since all the edges connecting v to Y ′′ have colour 2,we can extend the copy of H − w ⊆ Gnim

2 [Y ′′] to a copy of H by adding v. This contradicts thedefinition of Gnim

2 . Hence,

nim(ψ;H) = e(Gnim2 ) + e(Gnim

1 ) ≤ 3h

(10h

h

)n+ ex(n,H − w) + h2

< ex(n,H)− 22h2n+ 4h

(10h

h

)n < ex(n,H).

Thus, any colouring with NIM-edges of two different colours is not extremal.

3.2 General bipartite graphs

In this subsection, we will prove Theorems 1.12 and 1.13.

Proof of Theorem 1.12. As H contains a cycle, ex(n,H)/n→∞ as n→∞. Then by Lemma 3.2,we may assume that, for example, Gnim

1 is {K1,h,Mh/2}-free. Since Gnim2 is H-free, we immediately

get that

nim2(n,H) ≤ ex(n,H) + ex(n, {K1,h,Mh/2}) ≤ ex(n,H) + h2,

as desired.

Proof of Theorem 1.13. Let us first present the part of the proof which works for an arbitrarynumber of colours k and any forest T . Let h = v(T ).

The stated lower bound on nimk(n;T ) can be obtained by using the argument of Ma [33]. Fixsome maximum T -free graph H on [n] and take uniform independent permutations σ1, . . . , σk−1 of[n]. Iteratively, for i = 1, . . . , k − 1, let the colour-i graph Gi consists of those pairs {σi(x), σi(y)},xy ∈ E(H), that are still uncoloured. Finally, colour all remaining edges with colour k. Clearly, alledges of colours between 1 and k − 1 are NIM-edges. Since e(H) ≤ hn = O(n), the expected sizeof∑k−1

i=1 e(Gi) is at least

(k − 1)e(H)−(k − 1

2

)e(H)2

(n

2

)−1≥ (k − 1)ex(n, T )− k2h2.

By choosing the permutations for which∑k−1

i=1 e(Gi) is at least its expectation, we obtain therequired bound.

Let us turn to the upper bound. Fix an extremal G with colouring φ :([n]2

)→ [k], so

nimk(φ;T ) = nimk(n;T ). For every 1 ≤ i ≤ k, denote

Ai := {v ∈ V (G) : ∃u ∈ V (G), uv ∈ E(Gnimi )} and ai := |Ai|.

In other words, Ai is the set of all vertices incident with at least one i-coloured NIM-edge. Notethat

nimk(φ;T ) ≤k∑i=1

ex(ai, T ). (8)

12

Also, for every X ⊆ [k], define

BX := {v ∈ V (G) : v ∈ Ai ⇔ i ∈ X} = ∩i∈XAi \ (∪j /∈XAj) and bX := |BX |.

In other words, BX is the set of vertices which are incident with edges in Gnimi if and only if i ∈ X.

By definition, for two distinct subsets X,Y ⊆ [k], BX ∩BY = ∅.

Claim 3.8. For every two subsets X,Y ⊆ [k] with X ∪ Y = [k], min{bX , bY } < 6kh.

Proof of Claim. Assume on the contrary that there exist two subsets X,Y ⊆ [k] such that X∪Y =[k] and bX , bY ≥ 6kh. Let B′X ⊆ BX and B′Y ⊆ BY be such that |B′X | = |B′Y | = 6kh. Byaveraging, some colour, say colour 1, contains at least 1/k proportion of edges in G[B′X , B

′Y ]. Set

F = G1[B′X , B

′Y ]. Then there exists F ′ ⊆ F on vertex set B′′X∪B′′Y , where B′′X ⊆ B′X and B′′Y ⊆ B′Y ,

such that the minimum degree of F ′ is at least half of the average degree of F , that is,

δ(F ′) ≥ e(F )

|V (F )|≥

|B′X | · |B′Y |k · (|B′X |+ |B′Y |)

=(6kh)2

k · 12kh= 3h.

Let v ∈ V (T ) be a leaf, u be its only neighbour, and T ′ := T −v, where T −v is the forest obtainedfrom deleting the leaf v from T . Since X ∪ Y = [k], without loss of generality, we can assume that1 ∈ X. Fix an arbitrary vertex x ∈ B′′X and let w be a Gnim

1 -neighbour of x. (Such a vertex existsas x ∈ B′′X ⊆ BX and 1 ∈ X.) Then δ(F ′−w) ≥ δ(F ′)−1 ≥ 2h. We can then greedily embed T ′ inF ′−w with x playing the role of u. As this copy of T ′ is in F ′−w ⊆ G1, together with xw ∈ Gnim

1 ,we get a monochromatic copy of T with an edge in Gnim, a contradiction (see Figure 3).

���

�n �wg

T�

���

Y

xw

Figure 3: Finding a copy of T ∈ G1.

We will divide the rest of the proof into two cases.

Case 1: There exists a subset X ⊂ [k] such that |X| = k − 1 and bX ≥ 6kh.

Let {j} = [k]\X, and Y be the collection of all subsets of [k] containing j. By Claim 3.8, bY < 6kh,for every set Y ∈ Y, implying that aj =

∑Y ∈Y bY < 2k · 6kh. Hence, by (8),

nimk(φ;T ) ≤∑i∈[k]

ex(ai, T ) ≤∑

i∈[k]\{j}

ex(ai, T ) + ex(2k · 6kh, T )

≤ (k − 1)ex(n, T ) + 2k · 6kh2.

Thus the theorem holds in this case.

Case 2: For all subsets X ⊂ [k] with |X| = k − 1, we have bX < 6kh.

13

By Claim 3.8, we have b[k] ≤ 2 · 6kh. Hence, all but at most (k + 2)6kh vertices are adjacentto NIM-edges with at most k − 2 different colours, which implies that they are in at most k − 2different sets Ai. Therefore,

a1 + · · ·+ ak ≤ (k − 2)n+ 12(k + 2)kh. (9)

Now our analysis splits further, depending on the cases of Theorem 1.13. If k = 2, then we aredone by (8) and (9):

nim2(φ;T ) ≤ h(a1 + a2) ≤ 96h2 ≤ ex(n, T ).

Thus it remains to consider the case when k ≥ 3 and T is a tree. By taking the disjoint unionof two maximum T -free graphs, we see that the Turan function of T is superadditive, that is,

ex(`, T ) + ex(m,T ) ≤ ex(`+m,T ), for any `,m ∈ N. (10)

The Fekete Lemma implies that ex(m,T )/m tends to a limit τ . Since, for example, ex(m,T ) ≤ hm,we have that τ ≤ h, in particular, τ is finite. Also, excluding the case T = K2 when the theoremtrivially holds, we have τ > 0. In particular, | ex(m,T )/m − τ | < c for all large m, where c :=τ/(3k − 4) > 0 satisfies (τ + c)(k − 2) = (k − 1)(τ − 2c).

Thus (8), (9), (10) and the fact that n is sufficiently large give that

nimk(φ;T ) ≤ ex(a1 + · · ·+ ak, T ) ≤ ex((k − 2)n+ 12(k + 2)kh, T )

≤ (τ + c)((k − 2)n+ 12(k + 2)kh

)≤ (k − 1)(τ − 2c)n+ 24(k + 2)kh2

≤ (k − 1)(τ − c)n ≤ (k − 1)ex(n, T ).

This finishes the proof of Theorem 1.13.

4 Proofs of Theorems 1.5 and 1.6

We need the following lemma, which states that the reduced graph of the NIM-graph cannot havea large clique, linking the nim function to the new Ramsey variant r∗.

Lemma 4.1. For i ∈ [k], let Hi be a non-bipartite graph, and let 1/k, 1/r ≥ γ � ε � 1/N > 0,where r := R(a1 − 1, . . . , ak − 1) and ai := χ(Hi). Let V1, . . . , Vm be disjoint sets, each of size atleast N . Take any φ :

(V2

)→ [k], where V := V1 ∪ · · · ∪ Vm, and let Gnim be the NIM-graph of φ.

Then the graph R := R(ε, γ, φ|E(Gnim), (Vi)mi=1) is Kr∗+1-free, where r∗ := r∗(H1, . . . ,Hk).

Proof. Given the graphs Hi with ai = χ(Hi), and r = R(a1, . . . , ak), choose additional constantsso that the following hierarchy holds:

1

r� γ � ε1 �

1

M� ε� 1

N> 0.

Let the Vi’s and φ be as in the statement of the lemma. For each i ∈ [m], apply the MulticolourRegularity Lemma (Lemma 2.3) with constants ε1 and 1/ε1 to the k-coloured complete graph onVi to obtain an ε1-regular partition Vi = Ui,1 ∪ · · · ∪ Ui,mi with 1/ε1 ≤ mi ≤ M . Let Ri :=R(ε1, γ, φ|(Vi2 ), (Ui,j)

mij=1) be the associated reduced graph.

14

Note that the fraction of the elements xy ∈(Vi2

)with x ∈ Ui,a and y ∈ Ui,b such that the pair

(Ui,a, Ui,b) is not ε1-regular in some colour or satisfies a = b is at most ε1 + 1/mi. Since γ ≤ 1/k,

the remaining elements of(Vi2

)come from edges of Ri. Recall that mi = v(Ri). Thus, we have that

e(Ri) ≥(1− ε1 − 1/mi)

(|Vi|2

)d |Vi|/mi e2

≥ (1− 2ε1)

(mi

2

). (11)

Let ξ : E(R) → [k] be the colouring of R. We extend it to the vertices of R as follows. Takei ∈ [m]. Let ξi : E(Ri)→ [k] be the colouring of Ri. By (11) and since v(Ri) ≥ 1/ε1 and ε1 � 1/r,we have that e(Ri) > ex(mi,Kr). By Turan’s theorem [48], the graph Ri contains an r-clique. Bythe definition of r, the restriction of the k-edge-colouring ξi to this r-clique contains a colour-p copyof Kap−1 for some p ∈ [k]. Let ξ assign the colour p to Vi.

Suppose to the contrary that some (r∗+1)-set A spans a clique in R. The restriction of ξ to(A≤2)

violates either (P1) or (P2) from Definition 1.3. We will derive contradictions in both cases, thusfinishing the proof. If ξ contains an edge-monochromatic homomorphic copy of some Hi in colouri ∈ [k], then by the Embedding Lemma (Lemma 2.4) the colour-i subgraph of Gnim contains a copyof Hi, a contradiction to Gnim consisting of the NIM-edges. So suppose that (P2) fails, say, somepair ViVj ∈

(A2

)satisfies ξ(ViVj) = ξ(Vi), call this colour p. By the definition of ξ(Vi), Ri contains

an (ap − 1)-clique of colour p under ξi, say with vertices U1, . . . , Uap−1 ∈ V (Ri). Observe thatε1 ≥ max{2ε, εM} ≥ max{2ε, ε · v(Ri)} and p is the majority colour on edges in Gnim[Vi, Vj ]. TheSlicing Lemma (Lemma 2.5) with e.g. α := 1/M gives that each pair (Vj , Uh) with h ∈ [ap − 1] is(ε1, γ/2)-regular in Gnim

p . The Embedding Lemma (Lemma 2.4) gives a copy of Hp in G containing

at least one (in fact, at least δ(Hp)) edges of Gnim[Vi, Vj ], a contradiction.

Proof of Theorem 1.5. For the upper bound, let 1 � ε � γ � ε1 > 0. Let n be large andsuppose to the contrary that there exists some colouring φ : E(Kn)→ [k] that violates (4). Applythe Multicolour Regularity Lemma (Lemma 2.3) to the NIM-graph Gnim of φ with parameters ε1and 1/ε1. A calculation similar to the one in (11) applies here, where additionally one has todiscard at most kγ

(n2

)edges in NIM(φ) coming from pairs that have density less than γ in each

colour. By γ � ε, we conclude that the reduced graph R = R(ε1, γ, φ|E(Gnim)) of Gnim has at least

(1− 1/r∗ + ε/2)v(R)2

2 edges. By Turan’s theorem, Kr∗+1 ⊆ R, contradicting Lemma 4.1.

For the lower bound, take a feasible k-colouring ξ of([r∗]≤2), where r∗ := r∗(H1, . . . ,Hk). If

possible, among all such colourings take one such that all singletons have the same colour. Considerthe blow-up colouring φ := ξ(X1, . . . , Xr∗) where the sets Xi form an equipartition of [n].

Let us show that every edge of K[X1, . . . , Xr∗] is a NIM-edge. Take any copy F of Hi which isi-monochromatic in φ. Since the restriction of ξ to

([r∗]2

)has no homomorphic copy of Hi by (P1),

the graph F must use at least one edge that is inside some Vj . If ξ assigns the value i only tosingletons, then no edge of the colour-i graph F can be a cross-edge. Otherwise, if F is connected,then E(F ) ⊆

(Vj2

)because no edge between Vj and its complement can have φ-colour i by (P2).

We conclude that every cross-edge is a NIM-edge, giving the required lower bound.

Proof of Theorem 1.6. Choose ε � ε1 � δ � γ � ε2 � 1/n > 0. Let φ be as in the theorem.Apply the Regularity Lemma (Lemma 2.3) to NIM-graph Gnim with parameters ε2 and 1/ε2 toget an ε2-regular partition V (Gnim) = V1 ∪ · · · ∪ Vm. Let R = R(ε2, γ, φ|E(Gnim), (Vi)

mi=1) be the

reduced graph. A similar calculation as in (11) yields that e(R) ≥(1− 1

r∗ − 2δ)m2

2 . On the otherhand, by Lemma 4.1, R is Kr∗+1-free. Thus, the Erdos-Simonovits Stability Theorem [11, 42]

15

implies that δedit(R, T (m, r∗)) ≤ ε1m2/2. Let a partition V (R) = U1 ∪ · · · ∪ Ur∗ minimise |E(R)4E(K[U1, . . . ,Ur∗ ])|. We know that the minimum is at most ε1m

2/2. Let V (Gnim) = W1 ∪ · · · ∪Wr∗

be the partition induced by Ui’s, i.e., Wi := ∪Vj∈UiVj for i ∈ [r∗]. Let G′ be the graph obtainedfrom Gnim by removing all edges that lie in any cluster Vi; or between those parts Vi and Vj suchthat ViVj is not an ε2-regular pair or belongs to E(R)4 E(K[U1, . . . ,Ur∗ ]). We have

|E(Gnim)4E(G′)| = |E(Gnim)\E(G′)| ≤ m· (n/m)2

2+ε2m

2· n2

m2+|E(R)4E(T (m, r∗))|· n

2

m2≤ ε1n2.

As e(Gnim) ≥ (1− 1/r∗)n2/2− δn2, we have e(G′) ≥ (1− 1/r∗)n2/2− 2ε1n2. Since G′ is r∗-partite

(with parts W1, . . . ,Wr∗), a direct calculation gives that δedit(G′, T (n, r∗)) ≤ εn2/2. Finally, we

obtain

δedit(Gnim, T (n, r∗)) ≤ |E(Gnim)4 E(G′)|+ δedit(G

′, T (n, r∗)) ≤ ε1n2 +εn2

2≤ εn2,

as desired.

5 Proof of Theorem 1.7

The following lemma will be useful in the forthcoming proof of Theorem 1.7. It is proved by aneasy modification of the standard proof of Ramsey’s theorem.

Lemma 5.1 (Partite Ramsey Lemma). For every triple of integers k, r, u ∈ N there is ρ = ρ(k, r, u)such that if φ is a k-edge-colouring of the complete graph on Y1 ∪ · · · ∪ Yr, where Y1, . . . , Yr aredisjoint ρ-sets, then there are u-sets Ui ⊆ Yi, i ∈ [r], and ξ :

( [r]≤2)→ [k] such that ξ(U1, . . . , Ur) ⊆ φ.

(In other words, we require that each(Ui2

)and each bipartite graph [Ui, Uj ] is monochromatic.)

Proof. We use induction on r with the case r = 1 being the classical Ramsey theorem. Let r ≥ 2and set N := (u − 1)kr + 1. We claim that ρ := (2k)N ρ(k, r − 1, u) suffices here. Let ξ be anarbitrary k-edge-colouring of the complete graph on Y1 ∪ · · · ∪ Yr where each |Yi| = ρ.

Informally speaking, we iteratively pick vertices x1, . . . , xN in Yr shrinking the parts so thateach new vertex xi is monochromatic to each part. Namely, we initially let U0

i := Yi for i ∈ [r].Then for i = 1, . . . , N we repeat the following step. Given vertices x1, . . . , xi−1 and sets U i−1r ⊆Yr \ {x1, . . . , xi−1} and U i−1j ⊆ Yj for j ∈ [r − 1], we let xi be an arbitrary vertex of U i−1r and, for

j ∈ [r], let U ij be a maximum subset of U i−1j such that all pairs between xi and U ij have the same

colour, which we denote by cij ∈ [k]. Clearly, |U ij | ≥ (|U i−1j | − 1)/k (the −1 term is need for j = r),

which is at least (2k)N−i by a simple induction on i. Thus we can carry out all N steps. Moreover,each of the the final sets UN1 , . . . , U

Nr−1 has size at least ρ/(2k)N = ρ(k, r − 1, u). By the induction

assumption, we can find u-sets Uj ⊆ UNj , j ∈ [r−1], and ξ :([r−1]≤2)→ [k] with ξ(U1, . . . , Ur−1) ⊆ φ.

Each selected vertex xi comes with a colour sequence (ci1, . . . , cir) ∈ [k]r. So we can find a set

Ur ⊆ {x1, . . . , xN} of dN/kre = u vertices that have the same colour sequence (c1, . . . , cr). Clearly,all pairs in

(Ur

2

)(resp. [Ur, Uj ] for j ∈ [r− 1]) have the same colour cr (resp. cj). Thus if we extend

the colouring ξ to( [r]≤2)

by letting ξ(i, r) := ci for i ∈ [r− 1] and ξ(r) := cr, then ξ(U1, . . . , Ur) ⊆ φ,as required.

The main step in proving Theorem 1.7 is given by the following lemma.

16

Lemma 5.2. Under the assumptions of Theorem 1.7, there is n0 such that if φ is an arbitraryk-edge-colouring of G := Kn with n ≥ n0, e(Gnim) ≥ t(n, r∗) and

δ(Gnim) ≥ δ(T (n, r∗)), (12)

where δ denotes the minimum degree, then Gnim ∼= T (n, r∗) (in particular, e(Gnim) = t(n, r∗)).

Proof. Let Hi, i ∈ [k], and r∗ be as in Theorem 1.7. So r∗ = r∗(H1, . . . ,Hk). Let

N := maxi∈[k]

v(Hi)− 1 ≥ 1 and 1� ε� ε1 � 1/n0 > 0.

Let n ≥ n0 and let φ be an arbitrary k-edge-colouring of G := Kn.Let P = {V1, . . . , Vr∗} be a max-cut r∗-partition of Gnim. In particular, for every i, j ∈ [r∗] and

every v ∈ Vi, we have dGnim(v, Vj) ≥ dGnim(v, Vi). By applying Theorem 1.6 to Gnim, we have

e(Gnim[P]) ≥ t(n, r∗)− ε1n2. (13)

A simple calculation shows that |Vi| = nr∗ ± 3

√ε1n for all i ∈ [r∗].

Claim 5.3. For every i ∈ [r∗] and v ∈ Vi, dGnim(v, Vi) ≤ εn.

Proof of Claim. Assume to the contrary that there exist i ∈ [r∗] and v ∈ Vi such that dGnim(v, Vi) >εn. For each j ∈ [r∗], as P is a max-cut, there exists a colour ` ∈ [k] such that

dGnim`

(v, Vj) ≥ dGnim(v, Vj)/k ≥ dGnim(v, Vi)/k ≥ εn/k =: m.

So, for j ∈ [r∗], let Zj ⊆ NGnim`

(v, Vj) be any subset of size m. We have

e(Gnim[Z1, . . . , Zr∗ ]) ≤ e(Gnim[V1, . . . , Vr∗ ])(13)

≤ ε1n2. (14)

Let ρ := ρ(k, r∗, N), where ρ is the function from the Partite Ramsey Lemma (Lemma 5.1). Fori ∈ [r∗], let Yi be a random ρ-subset of Zi, chosen uniformly and independently at random. By (14),the expected number of missing cross-edges in Gnim[Y1, . . . , Yr∗ ] is at most

ε1n2

((m− 1

ρ− 1

)/(mρ

))2

= ε1

(ρk

ε

)2

< 1.

Thus there is a choice of the ρ-sets Yi’s such that Gnim[Y1, . . . , Yr∗ ] has no missing cross-edges. By

the definition of ρ, there are N -sets U1 ⊆ Y1, . . . , Ur∗ ⊆ Yr∗ and a colouring ξ :([r∗]≤2)→ [k] such

that ξ(U1, . . . , Ur∗) ⊆ φ.Note that ξ is feasible. Indeed, if we have, for example, ξ(ij) = ξ(i) =: c, then by taking one

vertex of Uj and all N vertices of Ui we get a colour-c copy of KN+1. However, since N+1 ≥ v(Hc),every edge of this clique is in an Hc-subgraph, contradicting the fact that all pairs in the completebipartite graph K[Ui, Uj ] are NIM-edges.

Consequently, as (H1, . . . ,Hk) is nice, ξ must assign the same colour to all singletons, saycolour 1. By construction, the vertex v is monochromatic into each Zi ⊇ Ui. So we can takeξ′ :

([r∗+1]≤2

)→ [k] such that ξ′(U1, . . . , Ur∗ , {v}) ⊆ φ, where we additionally let ξ′(r∗ + 1) := 1.

As r∗ = r∗(H1, . . . ,Hk), the colouring ξ′ violates (P1) or (P2). This violation has to include

17

the vertex r∗ + 1 since the restriction of ξ′ to([r∗]≤2)

is the feasible colouring ξ. We cannot havei ∈ [r∗] with ξ′(i, r∗ + 1) = 1 because otherwise Ui ∪ {v} is an (N + 1)-clique coloured 1 underφ, a contradiction to all pairs between Zi ⊇ Ui and v being NIM-edges. Therefore, there existsan edge-monochromatic homomorphic copy of Hj of colour j, say F , with r∗ + 1 ∈ V (F ). By thedefinition of homomorphism-criticality, there exists a homomorphism g : V (Hj)→ V (F ) such that|g−1(r∗ + 1)| = 1. Therefore, we can find an edge-monochromatic copy of Hj in colour j, withg−1(r∗ + 1) mapped to v, and all the other vertices of Hj mapped to vertices in U1 ∪ . . . ∪ Ur∗ , acontradiction to all pairs between this set and v being NIM-edges.

We next show that all pairs inside a part get the same colour under φ.

Claim 5.4. For any p ∈ [r∗] and any u1u2, u3u4 ∈(Vp2

), we have φ(u1u2) = φ(u3u4).

Proof of Claim. Suppose on the contrary that u1, . . . , u4 ∈ Vp violate the claim. Without lossof generality, let p = r∗. Let U := {u1, . . . , u4}. By (12), Claim 5.3 and the fact that |Vr| =n/r± 3

√ε1 n, all but at most 2εn edges from any u ∈ Vr∗ to V \Vr∗ are NIM-edges. For i ∈ [r∗− 1]

(resp. i = r∗), define Zi ⊆ Vi to be a largest subset of ∩4j=1NGnim(uj , Vi) (resp. Vr∗ \ U) with thesame colour pattern to U , i.e., for all x, x′ ∈ Zi and j ∈ [4] we have φ(ujx) = φ(ujx

′). By thePigeonhole Principle, we have for i ∈ [r∗ − 1] that

|Zi| ≥| ∩4j=1 NGnim(uj , Vi)|

k4≥ |Vi| − 4 · 2εn

k4≥ n

2r∗k4.

Also, |Zr∗ | ≥ (|Vr∗ | − 4)/k4 ≥ n/(2r∗k4).Similarly to the calculation after (14), there are N -subsets Ui ⊆ Zi, i ∈ [r∗], such that φ

contains the blow-up ξ(U1, . . . , Ur∗) of some ξ :([r∗]≤2)→ [k]. As in the proof of Claim 5.3, ξ is

feasible and assigns the same colour, say 1, to all singletons. Since φ(u1u2) 6= φ(u3u4), assume thate.g. φ(u1u2) 6= 1.

We define the colouring ξ′ :([r∗+1]≤2

)→ [k] so that ξ′(U1, . . . , Ur∗−1, {u1}, {u2}) ⊆ φ, where

additionally we let both ξ′(r∗) and ξ′(r∗+1) be 1. Note that ξ′(r∗, r∗+1) = φ(u1u2). Also, observethat we do not directly use the part Ur∗ when defining ξ′: the role of this part was to guaranteethat ξ is monochromatic on all singletons. By the definition of r∗, the colouring ξ′ violates (P1)or (P2).

Suppose first that ξ′ violates (P2), that is there is a pair ij ∈([r∗+1]

2

)with ξ′(ij) = 1. Since

ξ′(r∗, r∗ + 1) = φ(u1u2) 6= 1, we have {i, j} 6= {r∗, r∗ + 1}. Also, we cannot have i, j ∈ [r∗ − 1],

because ξ′ coincides on([r∗−1]≤2

)with the feasible colouring ξ. So we can assume by symmetry that

i ∈ [r∗ − 1] and j = r∗. However, then the vertex u1 is connected by NIM-1-edges to the colour-1clique on the N -set Ui, a contradiction.

We may now assume that the colouring ξ′ violates (P1). Let this be witnessed by an edge-monochromatic homomorphic copy of Hj of colour j, say F . If F contains exactly one vertexfrom {r∗, r∗ + 1}, then by an argument similar to the last part of the proof of Claim 5.3 we get acontradiction. Otherwise, if {r∗, r∗+ 1} ⊆ V (F ), then, by the definition of homomorphism-critical,there exists a homomorphism g : V (Hj)→ V (F ) such that |g−1(r∗)| = |g−1(r∗+1)| = 1. Therefore,we can find an edge-monochromatic copy of Hj in colour j, with g−1(r∗) (resp. g−1(r∗+1)) mappedto u1 (resp. u2), and all the other vertices ofHj mapped to vertices in U1∪. . .∪Ur∗−1, a contradictionto all pairs between this set and {u1, u2} being NIM-edges.

18

Let i ∈ [r∗]. By Claim 5.4 we know that G[Vi] is a monochromatic clique. Since |Vi| ≥maxj∈[k] v(Hj), no pair inside Vi is a NIM-edge. Thus Gnim is r∗-partite. Our assumption e(Gnim) ≥t(n, r∗) implies that Gnim is isomorphic to T (n, r∗), as desired.

We are now ready to prove the desired exact result.

Proof of Theorem 1.7. We know by Theorem 1.5 that nim(n;H1, . . . ,Hk) ≥ t(n, r∗) for all n.On the other hand, let n0 be the constant returned by Lemma 5.2. Let n ≥ n20 and let ψ be

an extremal colouring of G := Kn. In order to finish the proof of the theorem it is enough to showthat necessarily Gnim ∼= T (n, r∗).

Initially, let i = n, Gn := G and φn := ψ. Iteratively repeat the following step as long aspossible: if the NIM-graph of φi has a vertex xi of degree smaller than δ(T (i, r∗)), let φi−1 be therestriction of φi to the edge-set of Gi−1 := Gi−xi and decrease i by 1. Suppose that this procedureends with Gm and φm.

Note that, for every i ∈ {m+ 1, . . . , n}, we have that

t(i− 1, r∗) = t(i, r∗)− δ(T (i, r∗)),

nim(φi−1) ≥ nim(φi)− δ(T (i, r∗)) + 1,

the latter inequality following from the fact that every NIM-edge of φi not incident to xi is neces-sarily a NIM-edge of φi−1. These two relations imply by induction that

nim(φi) ≥ t(i, r∗) + n− i, for i = n, n− 1, . . . ,m. (15)

In particular, it follows that m > n0 for otherwise NIM(φn0) is a graph of order n0 with atleast n − n0 >

(n0

2

)edges, which is impossible. Thus Lemma 5.2 applies to φm and gives that

NIM(φm) ∼= T (m, r∗). By (15) we conclude that m = n, finishing the proof of Theorem 1.7.

6 Proofs of Theorems 1.8 and 1.9

Next we will show that Conjecture 1.4 holds for the 3-colour case.

Proof of Theorem 1.8. Take an arbitrary feasible 3-colouring ξ of( [r]≤2), where r = R(a2, a3) − 1.

It suffices to show that ξ assigns the same colour to all the singletons in [r]. Indeed, suppose

that (Ka1 ,Ka2 ,Ka3) is not nice. Then there exists a feasible 3-colouring ξ∗ of([r∗]≤2)

that is notmonochromatic on the singletons in [r∗], where r∗ := r∗(Ka1 ,Ka2 ,Ka3) ≥ r. Up to relabeling,we may assume that [r] contains two singletons of different colours in ξ∗. We then arrive to acontradiction, as the restriction of ξ∗ on [r] is also feasible. Fix now an arbitrary feasible 3-

colouring of([r∗]≤2), which assigns the same colour, say colour i, to all the singletons in [r∗]. Then

due to (P2), colour i cannot appear on([r∗]2

), and so r∗ ≤ R(aj , ak)−1 ≤ r, where {j, k} = [3]\{i}.

For i ∈ [3], let Vi be the set of vertices with colour i. Thus we have a partition [r] = V1∪V2∪V3.For i, j ∈ [3], let ωj(Vi) be the size of the largest edge-monochromatic clique of colour j in Vi.

Observe the following properties that hold for every triple i, j, ` ∈ [3] of distinct indices, i.e., for{i, j, `} = [3]. By (P2), the colour of every edge inside Vi is either j or ` while all the edges goingbetween Vj and V` have colour i. By the latter property and (P1), we have

ωi(V`) + ωi(Vj) ≤ ai − 1 and Vj 6= ∅ ⇒ ωi(V`) ≤ ai − 2. (16)

19

For notational convenience, define r(n1, . . . , nk) := R(n1, . . . , nk) − 1 to be one less than theRamsey number (i.e. it is the maximum order of a clique admitting a (Kn1 , . . . ,Knk

)-free edge-colouring). By the definition of ωj(Vi), we also have

|Vi| ≤ r(ωj(Vi) + 1, ω`(Vi) + 1). (17)

Also, we will use the following trivial inequalities involving Ramsey numbers that hold for arbitraryintegers a, b, c ≥ 2: r(a, b) + r(a, c) ≤ r(a, b+ c− 1) and r(a, b) < r(a+ 1, b).

First, let us derive the contradiction from assuming that each colour i ∈ [3] appears on at leastone singleton, that is, each Vi is non-empty. In order to reduce the number of cases, we allow toswap colours 1 and 2 to ensure that ω1(V2) ≥ ω2(V1). Thus we do not stipulate now which of a1and a2 is larger. Observe that

|V1|+ |V2|(17)

≤ r(ω2(V1) + 1, ω3(V1) + 1) + r(ω1(V2) + 1, ω3(V2) + 1)

≤ r(ω1(V2) + 1, ω3(V1) + 1) + r(ω1(V2) + 1, ω3(V2) + 1)

≤ r(ω1(V2) + 1, ω3(V1) + ω3(V2) + 1)(16)

≤ r(ω1(V2) + 1, a3). (18)

Hence, we get

r = |V1|+ |V2|+ |V3|(17),(18)

≤ r(ω1(V2) + 1, a3) + r(ω1(V3) + 1, ω2(V3) + 1)

(16)

≤ r(ω1(V2) + 1, a3) + r(ω1(V3) + 1, a2 − 1)

< r(ω1(V2) + 1, a3) + r(ω1(V3) + 1, a3)

≤ r(ω1(V2) + ω1(V3) + 1, a3)(16)

≤ r(a1, a3) ≤ r.

The above contradiction shows that, for some ` ∈ [3], the part V` is empty. Let {i, j, `} = [3];thus [r] = Vi ∪ Vj . It remains to derive a contradiction by assuming that each of Vi and Vj isnon-empty. By the symmetry between i and j, we can assume that ωj(Vi) ≥ ωi(Vj). Then we have

r = |Vi|+ |Vj |(17)

≤ r(ωj(Vi) + 1, ω`(Vi) + 1) + r(ωi(Vj) + 1, ω`(Vj) + 1)

≤ r(ωj(Vi) + 1, ω`(Vi) + 1) + r(ωj(Vi) + 1, ω`(Vj) + 1)

≤ r(ωj(Vi) + 1, ω`(Vi) + ω`(Vj) + 1)

(16)

≤ r(aj − 1, a`) < r(aj , a`) ≤ r,

which is the desired contradiction that finishes the proof of Theorem 1.8.

Next, let us present the proof that r∗(3, 3, 3, 3) = 16, the only non-trivial 4-colour case that wecan solve.

Proof of Theorem 1.9. Let ξ :([16]≤2)→ [4] be an arbitrary feasible colouring. It is enough to show

that all singletons in [16] get the same colour. For every i ∈ [4], let Vi denote the set of verticesof colour i. Suppose there are at least two different colours on the vertices, say V3, V4 6= ∅. As 5does not divide 16, there exists at least one class, say V3, of size not divisible by 5, i.e., |V3| 6≡ 0(mod 5). Choose an arbitrary vertex v ∈ V4. Since ξ is a feasible colouring, by (P2) the edges

20

incident to v cannot have colour ξ(v) = 4. We can then partition [16] \ {v} = ∪j∈[3]Wj , where

Wj := {u : ξ(uv) = j}. Let j ∈ [3]. By (P1) and (P2), colour j is forbidden in(Wj

≤2). Then

by Theorem 1.8, |Wj | ≤ r∗(K3,K3,K3) = R(3, 3) − 1 = 5. Since∑

j∈[3] |Wj | = 15, we havethat |Wj | = 5 for every j ∈ [3]. Again by Theorem 1.8, all vertices in Wj should have the samecolour. Recall that v ∈ V4, so V3 ⊆ ∪j∈[3]Wj and consequently V3 is the union of some Wj ’s. Thiscontradicts |V3| 6≡ 0 (mod 5).

7 Concluding remarks

• As pointed out by a referee, the function nim(n;H1, . . . ,Hk) is related to that of exr(n;H1, . . . ,Hr),which is the maximum size of an n-vertex graph G that can be r-edge-coloured so that thei-th colour is Hi-free for all i ∈ [r]. Indeed, we have the following lower bound:

nim(n;H1, . . . ,Hk) ≥ maxi∈[k]

exk−1(H1, . . . ,Hi−1, Hi+1, . . . ,Hk).

It is not inconceivable that the equality holds above if n ≥ n0(H1, . . . ,Hk). Theorems 1.7and 1.11 give classes of instances, when we have equality above. We refer the readers toSection 5.3 of [25] for more on the function exr.

• The Ramsey variant r∗ introduced here is related to the version of Ramsey numbers studiedby Gyarfas, Lehel, Schelp and Tuza [24]. In particular, Proposition 5 in [24] states thatr∗(K3,K3,K3,K3) = 16, which is the consequence of the fact that (K3,K3,K3,K3) is nicefrom Theorem 1.9.

• We prove in Theorem 1.13 that for any tree T , nimk(n;T ) = (k − 1)ex(n, T ) + OT (1). LetT be an (h+ 1)-vertex tree and suppose that the Erdos-Sos conjecture holds, i.e. ex(n, T ) ≤(h − 1)n/2. Then for each n ≥ k2h2 with h|n, we can get rid of the additive error term inthe lower bound, namely, it holds that nimk(n;T ) ≥ (k − 1)ex(n, T ). This directly followsfrom known results on graph packings. We present here a short self-contained proof (with aworse bound on n). Let F be the disjoint union of n/h copies of Kh. Let fi : V (F ) → [n],i ∈ [k − 1], be k − 1 arbitrary injective maps and let Fi be the graph obtained by mappingF on [n] via fi. It suffices to show that we can modify fi’s to have E(Fi) ∩ E(Fj) = ∅ for

any ij ∈([k−1]2

). Indeed, then the lower bound is witnessed by colouring e ∈ E(Kn) with

colour-i if e ∈ E(Fi), for each i ∈ [k − 1], and with colour-k otherwise. Suppose that there isa “conflict” uv ∈ E(Fi) ∩ E(Fj). Let F ∗ := ∪i∈[k−1]Fi. Note that ∆(F ∗) ≤ (k − 1)(h − 1).As n > ∆(F ∗)2 + 1, there exists a vertex w that is at distance at least 3 from v. We claimthat switching v and w in fi will remove all conflicts at v and w. If true, one can thenrepeat this process till all conflits are removed to get the desired fi’s. Indeed, suppose thatafter switching v and w, there is a conflict wz ∈ E(Fi) ∩ E(F`) for some z ∈ NFi(v) and` ∈ [k − 1] \ {i}. Then w, z, v form a path of length 2 in F ∗, contradicting the choice of w.

It would be interesting to prove a matching upper bound, i.e. to show that

nimk(n;T ) = (k − 1)ex(n, T )

for every tree T and sufficiently large n. Note that equality above need not be true when T isa forest. Indeed, consider M2, the disjoint union of two edges. Recall that ex(n,M2) = n− 1.

21

For any k ≥ 3 and n ≥ 4k, we have that nimk(n;M2) = (k − 1)ex(n,M2)− 12(k − 1)(k − 2).

Indeed, for any k-edge-colouring φ of Kn, one colour class, say colour-1, has size at least(n2

)/k ≥ 2(n − 1). As every edge share endpoints with at most 2n − 4 other edges, we

see that every colour-1 edge is in a copy of M2. Thus, nim(φ) ≤ exk−1(n,M2, . . . ,M2) =∑k−2i=0 (n− 1− i) = (k − 1)ex(n,M2)− 1

2(k − 1)(k − 2), as desired.

Acknowledgement

We would like to thank the referees for pointing out a flaw in the original draft, and for their carefulreading, which has greatly improved the presentation of this paper.

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